Precalculus Examples

$\frac{x^{3} - x^{2} + 7 x}{x - 5}$

Step 1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$5$

$x^{3}$

$x^{2}$

$7 x$

$0$

Step 2

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x$ .

					$x^{2}$
	$x$	-	$5$		$x^{3}$	-	$x^{2}$	+	$7 x$	+	$0$

Step 3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$5$		$x^{3}$	-	$x^{2}$	+	$7 x$	+	$0$
			+	$x^{3}$	-	$5 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 5 x^{2}$

				$x^{2}$
$x$	-	$5$		$x^{3}$	-	$x^{2}$	+	$7 x$	+	$0$
			-	$x^{3}$	+	$5 x^{2}$

Step 5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$
$x$	-	$5$		$x^{3}$	-	$x^{2}$	+	$7 x$	+	$0$
			-	$x^{3}$	+	$5 x^{2}$
					+	$4 x^{2}$

Step 6

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$
$x$	-	$5$		$x^{3}$	-	$x^{2}$	+	$7 x$	+	$0$
			-	$x^{3}$	+	$5 x^{2}$
					+	$4 x^{2}$	+	$7 x$

Step 7

Divide the highest order term in the dividend $4 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$4 x$
$x$	-	$5$		$x^{3}$	-	$x^{2}$	+	$7 x$	+	$0$
			-	$x^{3}$	+	$5 x^{2}$
					+	$4 x^{2}$	+	$7 x$

Step 8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$4 x$
$x$	-	$5$		$x^{3}$	-	$x^{2}$	+	$7 x$	+	$0$
			-	$x^{3}$	+	$5 x^{2}$
					+	$4 x^{2}$	+	$7 x$
					+	$4 x^{2}$	-	$20 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{2} - 20 x$

				$x^{2}$	+	$4 x$
$x$	-	$5$		$x^{3}$	-	$x^{2}$	+	$7 x$	+	$0$
			-	$x^{3}$	+	$5 x^{2}$
					+	$4 x^{2}$	+	$7 x$
					-	$4 x^{2}$	+	$20 x$

Step 10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	+	$4 x$
$x$	-	$5$		$x^{3}$	-	$x^{2}$	+	$7 x$	+	$0$
			-	$x^{3}$	+	$5 x^{2}$
					+	$4 x^{2}$	+	$7 x$
					-	$4 x^{2}$	+	$20 x$
							+	$27 x$

Step 11

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$	+	$4 x$
$x$	-	$5$		$x^{3}$	-	$x^{2}$	+	$7 x$	+	$0$
			-	$x^{3}$	+	$5 x^{2}$
					+	$4 x^{2}$	+	$7 x$
					-	$4 x^{2}$	+	$20 x$
							+	$27 x$	+	$0$

Step 12

Divide the highest order term in the dividend $27 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$4 x$	+	$27$
$x$	-	$5$		$x^{3}$	-	$x^{2}$	+	$7 x$	+	$0$
			-	$x^{3}$	+	$5 x^{2}$
					+	$4 x^{2}$	+	$7 x$
					-	$4 x^{2}$	+	$20 x$
							+	$27 x$	+	$0$

Step 13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$4 x$	+	$27$
$x$	-	$5$		$x^{3}$	-	$x^{2}$	+	$7 x$	+	$0$
			-	$x^{3}$	+	$5 x^{2}$
					+	$4 x^{2}$	+	$7 x$
					-	$4 x^{2}$	+	$20 x$
							+	$27 x$	+	$0$
							+	$27 x$	-	$135$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $27 x - 135$

				$x^{2}$	+	$4 x$	+	$27$
$x$	-	$5$		$x^{3}$	-	$x^{2}$	+	$7 x$	+	$0$
			-	$x^{3}$	+	$5 x^{2}$
					+	$4 x^{2}$	+	$7 x$
					-	$4 x^{2}$	+	$20 x$
							+	$27 x$	+	$0$
							-	$27 x$	+	$135$

Step 15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	+	$4 x$	+	$27$
$x$	-	$5$		$x^{3}$	-	$x^{2}$	+	$7 x$	+	$0$
			-	$x^{3}$	+	$5 x^{2}$
					+	$4 x^{2}$	+	$7 x$
					-	$4 x^{2}$	+	$20 x$
							+	$27 x$	+	$0$
							-	$27 x$	+	$135$
									+	$135$

Step 16

The final answer is the quotient plus the remainder over the divisor.

$x^{2} + 4 x + 27 + \frac{135}{x - 5}$

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